Chow variety

In mathematics, and more particularly in the field of algebraic geometry, Chow coordinates are a generalization of Plücker coordinates, applying to m-dimensional algebraic varieties of degree d in , that is, n-dimensional projective space. They are named for Wei-Liang Chow.

A Chow variety is a variety whose points correspond to all cycles of a given projective space of given dimension and degree.

Definition

To define the Chow coordinates, take the intersection of an algebraic variety Z of degree d and dimension m by linear subspaces U of codimension m. When U is in general position, the intersection will be a finite set of d distinct points.

Then the coordinates of the d points of intersection are algebraic functions of the Plücker coordinates of U, and by taking a symmetric function of the algebraic functions, a homogeneous polynomial known as the Chow form (or Cayley form) of Z is obtained.

The Chow coordinates are then the coefficients of the Chow form. Chow coordinates can generate the smallest field of definition of a divisor. The Chow coordinates define a point in the projective space corresponding to all forms.

Chow variety

The closure of the possible Chow coordinates is called the Chow variety.

Relation to Hilbert scheme

The Hilbert scheme is a variant of the Chow varieties. There is always a map (called the cycle map)

from the Hilbert scheme to the Chow variety.

Chow quotient

a Chow quotient parametrizes closures of generic orbits. It is constructed as a closed subvariety of a Chow variety.

Kapranov's theorem says that the moduli space of stable genus-zero curves with n marked points is the Chow quotient of Grassmannian by the standard maximal torus.

See also

References

  • Chow, W.-L.; van der Waerden., B. L. (1937), "Zur algebraische Geometrie IX.", Math. Ann., 113: 692–704, doi:10.1007/BF01571660
  • W. V. D. Hodge and Daniel Pedoe, vol. II
  • Mikhail Kapranov, Chow quotients of Grassmannian, I.M. Gelfand Seminar Collection, 29–110, Adv. Soviet Math., 16, Part 2, Amer. Math. Soc., Providence, RI, 1993.
  • Kollár, János (1996), Rational Curves on Algebraic Varieties, Berlin, Heidelberg: Springer-Verlag
  • Kollár, János, "Chapter 1", Book on Moduli of Surfaces
  • Kulikov, Val.S. (2001) [1994], "Chow variety", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
  • Mumford, David; Fogarty, J.; Kirwan, Frances (1994). Geometric invariant theory. Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)]. 34 (3rd ed.). Berlin, New York: Springer-Verlag. ISBN 978-3-540-56963-3. MR 1304906.
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